×

zbMATH — the first resource for mathematics

Inequalities for elementary means. (Ungleichungen für elementare Mittelwerte.) (German) Zbl 0813.26009
Für die verschiedenen positiven Zahlen \(x\) und \(y\) seien \(G(x, y):= \sqrt{xy}\), \(A(x, y):= (x+ y)/2\), \(L(x, y):= (x- y)/(\ln x- \ln y)\) und \(I(x, y):= e^{-1}(x^ x/ y^ y)^{1/(x- y)}\). Gezeigt werden die Ungleichungen \[ L(x, y)< \sqrt{L(G(x, y)^ 2, A(x,y)^ 2)}< \sqrt{I(G(x, y)^ 2, A(x, y)^ 2)}< I(x, y). \] Die rechte Seite dieser Ungleichungskette sowie einige weitere Ungleichungen, werden unter Benutzung von Identitäten bewiesen.

MSC:
26D15 Inequalities for sums, series and integrals
Keywords:
inequalities; means
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Alzer, Aufgabe 987. Elem. Math.43, 93 (1988) und44, 83-84 (1989).
[2] E. B. Leach andM. C. Sholander, Extended mean values. Amer. Math. Monthly85, 84-90 (1978). · Zbl 0379.26012 · doi:10.2307/2321783
[3] E. B. Leach andM. C. Sholander, Extended mean values II. J. Anal. Math. Appl.92, 207-223 (1983). · Zbl 0517.26007 · doi:10.1016/0022-247X(83)90280-9
[4] D. S.Mitrinov?, Analytic inequalities. Berlin-Heidelberg-New York 1970.
[5] J. Sándor, On the identric and logarithmic means. Aequationes Math.40, 261-270 (1990). · Zbl 0717.26014 · doi:10.1007/BF02112299
[6] H.-J. Seiffert, Aufgabe P 877. Praxis Math.28, 53 und 313-314 (1986).
[7] H.-J. Seiffert, Comment on problem 1365. Math. Mag.65, 356 (1992). · doi:10.2307/2691251
[8] H.-J. Seiffert, Aufgabe 1081, Elem. Math.49, 38 (1994).
[9] K. B. Stolarsky, Generalizations of the logarithmic mean. Math. Mag.48, 87-92 (1975). · Zbl 0302.26003 · doi:10.2307/2689825
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.